CHAP, viii] MISCELLANEOUS RESULTS ON CONGRUENCES. 259 N. H. Abel158 proved that we can solve by radicals any abelian equation, i. e., one whose rdots are r, $(r), <f>2(r) =</>[$(r)],.;., where </> is a rational function. H. J. S. Smith159 concluded that when the roots of a congruence can be similarly expressed modulo p, its solution can evidently be reduced to the solution of binomial congruences, and the expressions for the roots of the corresponding equation may be interpreted as the roots of the congruence. For the special case of=l, this was done by Poinsot in 1813-20 in papers discussed in the chapter on primitive roots. M. Jenkins159a noted that all solutions of a*==l(mod x) are x= Un=Uiu2 .. .un, where HI is any divisor of any power of a—1; u2 any divisor prime to a—1, of any power of aui —1;...; un any divisor, prime to a1771-2—!, of any power of aUn~l— 1. For a*+l=0 (mod x), modify the preceding by taking odd, factors of a-fl instead of factors of a —1. J. J. Sylvester160 proved that if p is a prune and the congruence /(z)=0 (mod p) of degree n has n real roots and if the resultant of f(x) and g(x) is divisible by p, then g(x) = Q has at least one root in common with/(z)==0. There are exactly p — 1 real roots of xp~l=l (mod pj). A. S. Hathaway161 noted the known similarity between equations and congruences for a prime modulus. He162 made abstruse remarks on higher congruences. G. Frattini163 proved that x2 — Dy^\ and x*—Dy2^\ are each solvable when the modulus is a prime p>5 and D^O. If d — B2—AC^Q, then A£4+2#£22/+C2/2=A (mod p) is solvable since dx*+\C can be made congruent to a square and hence to (Cy~{-Bx2)2. Likewise for ax2+2&rc+c=i/4. A. Hurwitz164 discussed the congruence of fractions and the theory of the congruence of infinite series. If </>(z) =r0-|-r'1x+ ... +rnxn/n\-\-... and if \(/(x) is a similar series with the coefficients sn, then 0 and \f/ are called congruent modulo ra if and only if rn=sn (mod w) for n= 1, 2,.. .. G. Cordone165 treated the general quartic congruence for a prime modulus /i by means of a cubic resolvent. The method is similar to Euler's solution of a quartic equation as presented by Giudice in Peano's Rivista di Matematica, vol. 2. For the special case x*+6Hx2+Kz=Q (mod M)> set t = (jjL — l)/2, r2 = 9H2—K', then if K is a quadratic residue of /*, there are four rational roots or none according as ( —SH+fO's+l or not; but if K is a non-residue, there are two rational roots or none according as one of the congruences (-3Jff+r)'s+l, (-3ff-r)'s-l is satisfied or not. 168Jour. fur Math., 4, 1829, 131; Oeuvres, 1, 114. ""Report British Assoc. 1860, 120 seq., §66: Coll. M. Papers, 1, 141-5. 169aMath. Quest. Educ. Times, 6, 1866, 91-3. 160Amer. Jour. Math., 2, 1879, 360-1; Johns Hopkins University Circulars, 1, 1881, 131. Coll. Papers, 3, 320-1. 161 Johns Hopkins Univ. Circulars, 1, 1881, 97. 162Amer. Jour. Math., 6, 1884, 316-330. 183Rendiconti Reale Accad. Lincei, Rome, (4), 1, 1885, 140-2. 1MActa Mathematica, 19, 1895, 356. 16fiRendiconti Circolo Mat. di Palermo 9, 1895, 209-243.